Question

write down the equations for the ci (by hand) for a clamped cubic spline with zero derivatives at the end knots. do not solve the system, just write down the equations

Answer 1

The equations for the ci coefficients of a clamped cubic spline with zero derivatives at the end knots are:

`\[2c_1 + c_2 = (3(y_2 - y_1))/(h_1) - 3m_0\]`

`\[h_i c_(i-1) + 2(h_i + h_(i+1))c_i + h_(i+1)c_(i+1) = (3)/(h_i)(m_(i+1) - m_i) - (3)/(h_(i+1))(m_i - m_(i-1))\]`

`\[c_(n-1) + 2c_n = (3(m_n - m_(n-1)))/(h_n) - 3m_n\]`

Explanation:

The equations for the ci coefficients in a clamped cubic spline with zero derivatives at the end knots are derived from the conditions set for the spline. In this context, the first equation represents the conditions at the first knot, involving the second derivative at the boundary to be zero.

The second equation generalizes the relationship between the ci coefficients and the second derivatives (mi) of the spline function within the interior knots, maintaining the continuity of the second derivative and the natural spline condition. The last equation signifies the conditions at the nth knot, again ensuring the second derivative at the boundary to be zero.

These equations are a result of establishing continuous second derivatives across the knots and integrating the cubic spline interpolation equations within the given conditions of zero derivatives at the ends. This system of equations helps in determining the coefficients ci of the clamped cubic spline without explicitly solving for the mi values. ""